# 8: Sparse Matrices

- Page ID
- 34845

A **sparse matrix** is a matrix in which most of the entries are zero. Such matrices are very commonly encountered in finite-difference equations. For example, when we discretized the 1D Schrödinger wave equation with Dirichlet boundary conditions, we saw that the Hamiltonian matrix had the tridiagonal form

\[\mathbf{H} = -\frac{1}{2h^2} \begin{bmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \ddots & 1 \\ & & 1 & -2\end{bmatrix} + \begin{bmatrix}V_0 \\ & V_1 \\& & \ddots \\ & & & V_{N-1}\end{bmatrix}.\]

Hence, if there are \(N\) diagonalization points, the Hamiltonian matrix has a total of \(N^{2}\) entries, but only \(O(N)\) of these entries are non-zero.